MATEMATIQKI VESNIK

59 (2007), 135–142

UDK 517.977

originalni nauqni rad research paper

WEYL’S AND BROWDER’S THEOREM FOR AN ELEMENTARY OPERATOR

F. Lombarkia and A. Bachir

Abstract. Let H be a separable infinite dimensional complex Hilbert space and let B(H) denote the algebra of bounded operators on H into itself. The generalized derivation δ_{A,B} is defined by δ_{A,B}(X) = AX − XB. For pairs C = (A_{1}, A_{2}) and D = (B_{1}, B_{2}) of operators, we define the elementary operator Φ_{C,D} by Φ_{C,D}(X) = A_{1}XB_{1} − A_{2}XB_{2}. If A_{2} = B_{2} = I, we get the elementary operator Δ_{A_{1},B_{1}}(X) = A_{1}XB_{1} − X. Let d_{A,B} = δ_{A,B} or Δ_{A,B}. We prove that if A, B^{∗} are log-hyponormal, then f(d_{A,B}) satisfies (generalized) Weyl’s Theorem for each analytic function f on a neighborhood of σ(d_{A,B}), we also prove that f(Φ_{C,D}) satisfies Browder’s Theorem for each analytic function f on a neighborhood of σ(Φ_{C,D}).

1. Introduction

Let B(H) denote the algebra of bounded linear operators acting on infinite dimensional separable Hilbert space H. If A ∈ B(H) we shall write ker(A) and ran(A) for the null space and the range of A, respectively. By α(A) and β(A) we denote the dimension of the kernel of A and the codimension of the range of A, respectively. Also write σ(A), σ_{a}(A), iso σ(A) for the spectrum, approximate point spectrum and the set of the isolated points of the spectrum of A, respectively. If A ∈ B(H), we say that A has the single-valued extension property at λ_{0}, SVEP (for short), if for every open disk D_{λ_{0}} centered at λ_{0} the only analytic function f : D_{λ_{0}} → H which satisfies the equation (λI − A)f(λ) = 0 for all λ ∈ D_{λ_{0}} is the function f ≡ 0.

Let A be a bounded linear operator on a Hilbert space H and 0 < p ≤ 1. A is called a p-hyponormal operator if (AA^{∗})^{p} ≤ (A^{∗}A)^{p}. Especially, A is called a hyponormal operator if p = 1 and semi-hyponormal if p = \frac{1}{2}. A is called a log-hyponormal operator if A is invertible and log(AA^{∗}) ≤ log(A^{∗}A). Since log : (0, ∞) → (−∞, ∞) is operator monotone, every invertible p-hyponormal operator is log-hyponormal. But the converse is not true [7]. However it is interesting to regard log-hyponormal operators as 0-hyponormal operators [7, 21]. The idea of log-hyponormal operator is due to Ando [3] and the first paper in which loghyponormality appeared is [13]. See [3, 7, 21] for properties of log-hyponormal operators. An operator A ∈ B(H) has a unique polar decomposition A = U|A|, where |A| = (A^{∗}A)^{\frac{1}{2}} and U is a partial isometry. If A = U|A|, then the Aluthge transform of A is defined by Ã = |A|^{\frac{1}{2}} U|A|^{\frac{1}{2}} .

AMS Subject Classification: 47B47, 47A30,47B20

Keywords and phrases: Elementary Operators, p-hyponormal,log-hyponormal, Weyl’s Theorem, single valued extension property.

The class of operators A, B^{∗} such that ker(d_{A,B}) ⊆ ker(d_{A^{∗},B^{∗}}) is large and includes in particular the class of log-hyponormal operators [17]. It is well known that if A, B^{∗} are log-hyponormal operators, then ker(d_{A,B}) ⊆ ker(d_{A^{∗},B^{∗}}) and asc(d_{A,B}) ≤ 1; this implies that d_{A,B} has the single valued extension property and hence satisfies Browder’s Theorem [11]. Here asc(d_{A,B}) denote the ascent of d_{A,B}.

The plan of this paper is as follows. In section 2 we prove that if C = (A_{1}, A_{2}) and D = (B_{1}, B_{2}) are pairs of operators and A_{1}, A_{2}, B_{1}^{∗}, B_{2}^{∗} are log-hyponormal such that A_{1} doubly commutes with A_{2} and B_{1} doubly commutes with B_{2}, then asc(Φ_{C,D}) ≤ 1. In section 3 we shall prove that if A, B^{∗} ∈ B(H) are loghyponormal, then d_{A,B} is isoloid and the range of d_{A,B} −λ is closed for each isolated point λ in the spectrum of d_{A,B}. In section 4 we shall show that if A, B^{∗} ∈ B(H) are log-hyponormal, then the Weyl’s Theorem holds for f(d_{A,B}) for every analytic function f defined on a neighborhood U of σ(d_{A,B}). Finally we shall prove the Browder’s Theorem for the elementary operator Φ_{C,D}.

2. The ascent of an elementary operator

Recall that the finite ascent property implies SVEP. In the following we prove that if C = (A_{1}, A_{2}) and D = (B_{1}, B_{2}) are pairs of operators and A_{1}, A_{2}, B_{1}^{∗}, B_{2}^{∗} are log-hyponormal such that A_{1} doubly commutes with A_{2} and B_{1} doubly commutes with B_{2}, then asc(Φ_{C,D}) ≤ 1.

Lemma 2.1. Let A, B ∈ B(H) be log-hyponormal operators such that A doubly commutes with B. Then AB is log-hyponormal.

Proof. If A and B are log-hyponormal, then A and B are invertible and

log |A|^{2} ≥ log |A^{∗}|^{2}, log |B|^{2} ≥ log |B^{∗}|^{2}.

Since AB = BA and AB^{∗} = B^{∗}A it follows that |AB|^{2} = |A|^{2}|B|^{2} = |B|^{2}|A|^{2}, and

|(AB)^{∗}|^{2} = |A^{∗}|^{2}|B^{∗}|^{2} = |B^{∗}|^{2}|A^{∗}|^{2}.

Hence

log |AB|^{2} = lim_{p→0^{+}} \frac{(|AB|^{2})^{p} − 1}{p} = lim_{p→0^{+}} \frac{(|A|^{2})^{p}(|B|^{2})^{p} − 1}{p}

= lim_{p→0^{+}} \frac{((|A|^{2})^{p} − 1)((|B|^{2})^{p} − 1) + (|A|^{2})^{p} + (|B|^{2})^{p} − 2}{p}

= log |A|^{2} + log |B|^{2}.

Similarly we have log |(AB)^{∗}|^{2} = log |A^{∗}|^{2} + log |B^{∗}|^{2}. Hence AB is invertible and

log |AB|^{2} − log |(AB)^{∗}|^{2} = log |A|^{2} − log |A^{∗}|^{2} + log |B|^{2} − log |B^{∗}|^{2} ≥ 0.

Thus AB is log-hyponormal.

Lemma 2.2. If A and B^{∗} are log-hyponormal, then asc(Δ_{A,B}) ≤ 1.

Proof. It is known that if A and B^{∗} are log-hyponormal, then ker(d_{A,B}) ⊆ ker(d_{A^{∗},B^{∗}}) [17] and this by [10] implies that ker(Δ_{A,B}) ⊆ ker(Δ_{A^{∗},B^{∗}}) and the result follows by [10].

Theorem 2.3. Let C = (A_{1}, A_{2}), D = (B_{1}, B_{2}) be pairs of operators in B(H). If A_{1}, B_{1}^{∗}, A_{2}, B_{2}^{∗} are log-hyponormal operators such that A_{1} doubly commutes with A_{2} and B_{1} doubly commutes with B_{2}, then asc(Φ_{C,D}) ≤ 1.

Proof. We have

Φ_{C,D} = A_{2}(Δ_{(A_{2}^{−1}A_{1}),(B_{1}B_{2}^{−1})})B_{2}

Since A_{2}^{−1} and (B_{2}^{−1})^{∗} = (B_{2}^{∗})^{−1} are log-hyponormal [3, Lemma 1.1] and A_{2}^{−1}A_{1} and B_{1}B_{2}^{−1} are log-hyponormal by Lemma 2.1, then by applying Lemma 2.2 we obtain

asc(Δ_{(A_{2}^{−1}A_{1}),(B_{1}B_{2}^{−1})}) ≤ 1.

Since

ker Φ_{C,D}^{n} = A_{2}^{n}(ker Δ_{(A_{2}^{−1}A_{1}),(B_{1}B_{2}^{−1})}^{n})B_{2}^{n}
= A_{2}(ker Δ_{(A_{2}^{−1}A_{1}),(B_{1}B_{2}^{−1})})B_{2} = ker Φ_{C,D},

it is asc(Φ_{C,D}) ≤ 1.

Corollary 2.4. Let C = (A_{1}, A_{2}), D = (B_{1}, B_{2}) be pairs of operators in B(H). If A_{1}, B_{1}^{∗}, A_{2}, B_{2}^{∗} are log-hyponormal operators such that A_{1} doubly commutes with A_{2} and B_{1} doubly commutes with B_{2}, then Φ_{C,D} has the single valued extension property.

Proof. The proof follows from Theorem 2.3 and [18, Proposition 1.8].

Corollary 2.5. Let C = (A_{1}, A_{2}), D = (B_{1}, B_{2}) be pairs of operators in B(H) and A_{1}, B_{1}^{∗}, A_{2}, B_{2}^{∗} be log-hyponormal operators such that A_{1} doubly commutes with A_{2} and B_{1} doubly commutes with B_{2}. A necessary and sufficient condition for ran(Φ_{C,D}) to be closed is that ran(Φ_{C,D}) + ker Φ_{C,D} is closed.

Proof. Since asc(Φ_{C,D}) ≤ 1, the proof follows from [18, Proposition 4.10.4].

3. The range of an elementary operator

Recall that A ∈ B(H) is said to be isoloid if λ ∈ iso σ(A) implies λ ∈ σ_{p}(A). In this section, we prove that if A and B^{∗} are log-hyponormal, then d_{A,B} is isoloid and ran(d_{A,B} −λ) is closed for each λ ∈ iso σ(d_{A,B}). We denote by L_{A} the operator of left multiplication by A and by R_{B} the operator of right multiplication by B.

Lemma 3.1. If A = U|A| is the polar decomposition of A and Ã = |A|^{\frac{1}{2}} U|A|^{\frac{1}{2}} its Aluthge transform, then

|A|^{\frac{1}{2}} A = Ã|A|^{\frac{1}{2}} and A(U|A|^{\frac{1}{2}} ) = (U|A|^{\frac{1}{2}} )Ã.

Lemma 3.2. Let A, B ∈ B(H). If A, B are invertible, then L_{A}R_{B} is invertible.

Proof. Since σ(L_{A}R_{B}) = ⋃{σ(zA) : z ∈ σ(B)} by [12, Theorem 3.2] we deduce that L_{A}R_{B} is invertible.

It is well known that d_{A,B} is isoloid when A and B^{∗} are hyponormal [11, Theorem 2.7]. The following theorem says that d_{A,B} retains this property in the case in which A and B^{∗} are log-hyponormal.

Theorem 3.3. If A and B^{∗} are log-hyponormal, then d_{A,B} is an isoloid.

Proof. The case d_{A,B} = Δ_{A,B}.

Let λ ∈ iso σ(Δ_{A,B}) such that λ ≠ −1, then (Δ_{A,B} −λ)(X) = AXB −(1+λ)X and it follows from [12, Theorem 3.2] that σ(Δ_{A,B} −λ) = ⋃{σ((−(1+λ)+zA) : z ∈

σ(B)}. Since iso σ(A) = iso σ(Ã) = iso σ(˜Ã) and iso σ(B) = iso σ(B̃) = iso σ(˜B̃) by [2, Corollary 2.3], we deduce that iso σ(Δ_{A,B}) = iso σ(Δ_{_{A},_{B}}) = iso σ(Δ_{˜_{A},˜_{B}}). The

operators A and B^{∗} being log-hyponormal, it follows from [21] that Ã (resp. B̃^{∗})

is semi-hyponormal and from [2] that ˜ Ã (resp. ˜ B̃^{∗}) is invertible and hyponormal, and the result follows from [11, Theorem 2.7] since σ_{p}(Δ_{˜_{A},˜_{B}}) = σ_{p}(Δ_{A,B}).

Let λ = −1,
λ ∈ iso σ(Δ_{A,B}) ⇒ 0 ∈ iso σ(Δ_{A,B} − λ) ⇒ 0 ∈ iso σ(L_{A}R_{B}).

This is a contradiction with Lemma 3.2.

The case d_{A,B} = δ_{A,B}.

Let λ ∈ iso σ(δ_{A,B}). Then 0 ∈ iso σ(δ_{A,B} − λ), where σ(δ_{A,B} − λ) = σ(A) − σ(B − λ) [12]. Hence iso σ(δ_{A,B}) = iso σ(δ_{_{A},_{B}}) = iso σ(δ_{˜_{A},˜_{B}}). The same arguments

cited above guarantees that σ_{p}(Δ_{˜_{A},˜_{B}}) = σ_{p}(Δ_{A,B}).

Remark 3.4. Note that in the above theorem we utilize the fact that Ã^{∗} is p-hyponormal if and only if (Ã)^{∗} p-hyponormal.

Theorem 3.5. If A, B^{∗} are log-hyponormal, then d_{A,B} − λ has closed range for each λ ∈ iso σ (d_{A,B}).

Proof. Let d_{A,B} = Δ_{A,B} and λ ∈ iso σ(Δ_{A,B}) such that λ ≠ −1. Let Y ∈ \overline{ran(Δ_{_{A},_{B}} − λ)} for all λ ∈ iso σ(Δ_{_{A},_{B}}). Then there exists (X_{n}) in B(H) such that

ÃX_{n}B̃ − (1 + λ) X_{n} ⟶ Y.

Let B̃ = Ũ|B̃| be the polar decomposition of B̃. Then

|Ã|^{\frac{1}{2}} ÃX_{n}B̃Ũ|B̃|^{\frac{1}{2}} − (1 + λ)|Ã|^{\frac{1}{2}} X_{n}Ũ|B̃|^{\frac{1}{2}} ⟶ |Ã|^{\frac{1}{2}} Y Ũ|B̃|^{\frac{1}{2}} .

From Lemma 3.1 we have |Ã|^{\frac{1}{2}} Ã = ˜ Ã|Ã|^{\frac{1}{2}} and B̃(Ũ|B̃|^{\frac{1}{2}} ) = (Ũ|B̃|^{\frac{1}{2}} )˜B̃. Hence

˜Ã|Ã|^{\frac{1}{2}} X_{n}B̃Ũ|B̃|^{\frac{1}{2}} − (1 + λ)|Ã|^{\frac{1}{2}} X_{n}Ũ|B̃|^{\frac{1}{2}} ⟶ |Ã|^{\frac{1}{2}} Y Ũ|B̃|^{\frac{1}{2}}
˜Ã|Ã|^{\frac{1}{2}} X_{n}Ũ|B̃|^{\frac{1}{2}} ˜ B̃ − (1 + λ)|Ã|^{\frac{1}{2}} X_{n}Ũ|B̃|^{\frac{1}{2}} ⟶ |Ã|^{\frac{1}{2}} Y Ũ|B̃|^{\frac{1}{2}} .

Since the operators A and B^{∗} are log-hyponormal, it follows from [21] that Ã

(resp. B̃^{∗}) is semi-hyponormal and from [2] that ˜ Ã (resp. ˜ B̃^{∗}) is invertible and hyponormal, and so from [11, Theorem 2.7] (Δ_{˜_{A},˜_{B}} − λ) has closed range for each

λ ∈ iso σ(Δ_{˜_{A},˜_{B}}) = iso σ(Δ_{_{A},_{B}}). Hence there exists Z ∈ B(H) such that

˜Ã|Ã|^{\frac{1}{2}} X_{n}Ũ|B̃|^{\frac{1}{2}} ˜ B̃ − (1 + λ)|Ã|^{\frac{1}{2}} X_{n}Ũ|B̃|^{\frac{1}{2}} ⟶ ˜ ÃZ ˜B̃ − (1 + λ)Z.

Since |Ã|^{\frac{1}{2}} and Ũ|B̃|^{\frac{1}{2}} are invertible, from Lemma 3.2 L_{|_{A}|^{\frac{1}{2}}} R_{_{U}|_{B}|^{\frac{1}{2}}} is invertible.

So there exists X ∈ B(H) such that Z = |Ã|^{\frac{1}{2}} XŨ|B̃|^{\frac{1}{2}} . Hence

˜ÃZ ˜B̃ − (1 + λ)Z = ˜ Ã|Ã|^{\frac{1}{2}} XŨ|B̃|^{\frac{1}{2}} ˜ B̃ − (1 + λ)|Ã|^{\frac{1}{2}} XŨ|B̃|^{\frac{1}{2}} .

From the uniqueness of the limit we obtain

˜Ã|Ã|^{\frac{1}{2}} XŨ|B̃|^{\frac{1}{2}} ˜ B̃ − (1 + λ)|Ã|^{\frac{1}{2}} XŨ|B̃|^{\frac{1}{2}} = |Ã|^{\frac{1}{2}} Y Ũ|B̃|^{\frac{1}{2}}
|Ã|^{\frac{1}{2}} ÃXB̃Ũ|B̃|^{\frac{1}{2}} − (1 + λ)|Ã|^{\frac{1}{2}} XŨ|B̃|^{\frac{1}{2}} = |Ã|^{\frac{1}{2}} Y Ũ|B̃|^{\frac{1}{2}} .

Then ÃXB̃ − (1 + λ)X = Y, and hence Y ∈ ran(Δ_{_{A},_{B}} − λ), and thus Δ_{_{A},_{B}} − λ has closed range for λ ∈ iso σ(Δ_{_{A},_{B}}). The same argument implies that (Δ_{A,B} − λ)has closed range for each λ ∈ iso σ(Δ_{A,B}) such that λ ≠ −1.

The case d_{A,B} = δ_{A,B}.

Let λ ∈ iso σ(δ_{A,B}). Then 0 ∈ iso σ(δ_{A,B} − λ). The same arguments implies that (δ_{A,B} − λ) has closed range for each λ ∈ iso σ(δ_{A,B}).

4. Weyl’s and Browder’s Theorem

An operator A ∈ B(H) is called Fredholm if it has closed range, finite dimensional null space, and its range has finite co-dimension. The index of a Fredholm operator is given by

I(A) := α(A) − β(A).

A bounded operator A is said to be a Weyl operator if it is Fredholm of index 0. Recall that the ascent of an operator A, denoted by asc(A), is the smallest nonnegative integer n such that ker(A^{n}) = ker(A^{n+1}). Analogously, the descent of an operator A, denoted by des(A), is the the smallest nonnegative integer n such that ran(A^{n}) = ran(A^{n+1}). It is well known that if asc(A) and des(A) are both finite then asc(A) = des(A) [16]. A ∈ B(H) is said to be a Browder operator if A is Fredholm with asc(A) = des(A) < ∞. Note that if A is Browder then A is Weyl, (see [15]). The essential spectrum σ_{e}(A), the Weyl spectrum σ_{w}(A) and the Browder spectrum σ_{b}(A) are defined by (see [14])

σ_{e}(A) : = { λ ∈ C : A − λ is not Fredholm }
σ_{w}(A) : = { λ ∈ C : A − λ is not Weyl }
σ_{b}(A) : = { λ ∈ C : A − λ is not Browder }.

Let σ_{0}(A) denote the set of Riesz points of A and σ_{00}(A) = { λ ∈ iso σ(A) : 0 < dim ker A < ∞ }. Then

iso σ(A) \ σ_{e}(A) = iso σ(A) \ σ_{w}(A) = σ_{0}(A) ⊆ σ_{00}(A).

Note that A ∈ B(H) satisfies Weyl’s Theorem (resp. Browder’s Theorem) if σ_{w}(A) = σ(A) \ σ_{00}(A) (resp. σ_{w}(A) = σ(A) \ σ_{0}(A)). A generalization of these notions are given in [4]; precisely, A ∈ B(H) is said to be generalized Fredholm or B-Fredholm, if there exists a positive integer n for which the induced operator A_{n} : ran(A^{n}) → ran(A^{n}) is Fredholm in the usual sense, and generalized Weyl, if in addition A_{n} has index zero. The generalized Weyl’s spectrum σ_{Bw}(A) of A is defined to be the set

{ λ ∈ C : (A − λ) is not generalized Weyl },

and we say that A satisfies generalized Weyl’s Theorem if σ_{Bw}(A) = σ(A) \ E(A), where E(A) is the set of all isolated eigenvalues of A. Note that if A satisfies generalized Weyl’s Theorem then A satisfies generalized Browder’s Theorem, see [4, Corollary 2.6]. Moreover, in [6] it is shown that if A satisfies generalized Weyl’s Theorem, then A satisfies Weyl’s Theorem, but the reverse implication in general fails [4, Example 4.1], and if A satisfies generalized Browder’s Theorem, then A satisfies Browder’s Theorem.

Let A, B^{∗} ∈ B(H) be log-hyponormal. Then the SVEP of d_{A,B} implies that the Browder’s Theorem holds for d_{A,B}. Recall [9, Theorem 2.5] that if an operator A ∈ B(H) has SVEP, then A satisfies Weyl’s Theorem if and only if ran (A − λ) is closed for every λ ∈ σ_{00} (A). Hence in view of Theorem 3.2, d_{A,B} satisfies Weyl’s Theorem.

Theorem 4.1. Let A, B^{∗} ∈ B(H) be log-hyponormal. If f is analytic on a neighbourhood of σ (d_{A,B}), then f (d_{A,B}) satisfies Weyl’s Theorem.

Proof. SVEP being stable under the functional calculus [18], d_{A,B} has SVEP ⟹ f (d_{A,B}) has SVEP for each f analytic in a neighbourhood of σ (d_{A,B}). This implies that σ_{b} (f (d_{A,B})) = σ_{w} (f (d_{A,B})) [14]. Since the spectral mapping theorem holds for σ_{b}, we have

σ_{w} (f (d_{A,B})) = σ_{b} (f (d_{A,B})) = f (σ_{b} (d_{A,B})) = f (σ_{w} (d_{A,B})) .

To complete the proof we have to show that

f (σ_{w} (d_{A,B})) = σ (f (d_{A,B})) ⧹σ_{00} (f (d_{A,B})) .

This follows from Theorem 3.1 and a limit argument applied to [20, Proposition 1].

Theorem 4.2. Let C = (A_{1}, A_{2}) , D = (B_{1}, B_{2}) be pairs of operators in B(H) and A_{1}, B_{1}^{∗}, A_{2}, B_{2}^{∗} be log-hyponormal operators such that A_{1} doubly commutes with A_{2} and B_{1} doubly commutes with B_{2}. If f is analytic on a neighborhood of σ (Φ_{C,D}), then f (Φ_{C,D}) satisfies Browder’s Theorem.

Proof. From Corollary 2.2 Φ_{C,D} has the single valued extension property. This implies that Φ_{C,D} satisfies Browder’s Theorem [10]. SVEP being stable under the functional calculus [18], Φ_{C,D} has SVEP ⟹ f (Φ_{C,D}) has SVEP for each f analytic in a neighborhood of σ (Φ_{C,D}). Hence f (Φ_{C,D}) satisfies Browder’s Theorem.

The operator A is said to be Drazin invertible if there is an operator T and a nonnegative integer n ∈ N such that

A^{n}T A = A^{n}, T AT = T and T A = AT.

It is known that A is Drazin invertible if and only if both asc(A) and des(A) are finite [19].

Theorem 4.3. Let A, B^{∗} ∈ B(H) be log-hyponormal. If f is analytic on a neighborhood of σ (d_{A,B}), then f (d_{A,B}) satisfies generalized Weyl’s Theorem.

Proof. Let λ ∈ σ (d_{A,B}) \ σ_{Bw} (d_{A,B}). Since d_{A,B} has SVEP, it follows upon arguing as in the proof of [4, Theorem 3.12] and an application of Theorem 3.1 that λ ∈ iso σ (d_{A,B}) = E (d_{A,B}). Conversely, if λ ∈ E (d_{A,B}), then d_{A,B} − λ is Fredholm of index 0 by Theorem 3.1 and Theorem 3.2. Hence d_{A,B} satisfies generalized Weyl’s Theorem. Let f be analytic on a neighborhood of σ (d_{A,B}), and let σ_{D} (d_{A,B}) = {λ ∈ C : (d_{A,B} − λ) is not Drazin invertible } denote the Drazin spectrum of d_{A,B}. Then σ_{D} (f (d_{A,B})) = f (σ_{D} (d_{A,B})) [4, Corollary 2.4]. Since d_{A,B} and f (d_{A,B}) have SVEP, σ_{D} (d_{A,B}) = σ_{Bw} (d_{A,B}) and σ_{D} (f (d_{A,B})) = σ_{Bw} (f (d_{A,B})) [4, Theorem 3.12]. Hence

f (σ_{Bw} (d_{A,B})) = f (σ (d_{A,B}) ⧹E (d_{A,B})) = σ_{Bw} (f (d_{A,B})) .

The isoloid property of d_{A,B}, Theorem 3.1 implies that

σ_{Bw} (f (d_{A,B})) = σ (f (d_{A,B})) ⧹E (f (d_{A,B}))

[5, Lemma 2.9], and the proof is complete.

Acknowledgment. The authors would to thank the referee for his thorough reading of the paper, his valuable suggestions, critical remarks, and incisive comments.

REFERENCES

[1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, w-hyponormal operators, Integral Equations Operator Theory, 13 (1990), 307–315.

[2] A. Aluthge and D. Wang, w-hyponormal operators, Integral Equations Operator Theory, 36 (2000), 1–10.

[3] T. Ando, Operators with a norm condition, Acta. Sci. Math. (Szeged), 33 (1972), 169–178.

[4] M. Berkani, B-Weyl spectrum and poles of the resolvent, J. Math. Anal. Appli., 272 (2002), 596–603.

[5] M. Berkani and A. Arroud, Generalized Weyl’s theorem and hyponormal operators, J. Aust. Math. Soc., 76 (2004), 291–302.

[6] M. Berkani and J. Koliha, Weyl type theorems for bounded linear operators, Acta. Sci. Math. (Szeged), 69 (2003), 359–376.

[7] M. Cho and K. Tanahashi, Isolated point of spectrum of p-hyponormal operators, loghyponormal opertors, pre-print.

[8] R.E. Curto and Y.M. Han, Weyl’s theorem, a-Weyl’s theorem and local spectral theory, J. London Math. Soc. 67 (2003), 499–509.

[9] B.P. Duggal, A remark on generalized Putnam-Fuglede theorem, Proc. Amer. Math. Soc. 129 (2000), 83–87.

[10] B.P. Duggal and S.V. Djordjević, Dunford’s property (C) and Weyl’s Theorem, Integral Equations Operator Theory, 43 (2002), 290–297.

[11] B.P. Duggal, Weyl’s Theorem for generalized derivation and an elementary operator, Math. Vesnik 54 (2002), 71–81.

[12] M.R. Embry and M. Rosemblum, Spectra, tensor product and linear operator equation, Pac. J. Math. 53 (1974) 95–107.

[13] M. Fujii, C. Hemeji and A. Matsumoto, Theorems of Ando and Saito for p-hyponormal operators, Math. Japonica 39 (1994), 595–598.

[14] R.E. Harte, Fredholm, Weyl and Browder theory, Proc. Irish Acad. A 85(2) (1985), 151–176.

[15] R.E. Harte and W.Y. Lee, Another note on Weyl’s theorem, Trans. Amer. Math. Soc. 349 (1997), 2115–2124.

[16] H. Heuser, Functional Analysis, Marcel Dekker, New York, 1982.

[17] I.H. Jeon, K. Tanahashi and A. Uchiyama, On quasisimilarity for log-hyponormal operators, Glasgow Math. J., 46 (2004), 169–176.

[18] K.B. Laursen, Operators with finite ascent, Pacific J. Math. 152, 2( 1992), 323–336.

[19] D.C. Lay, Spectral analysis using ascent, descent, nullity and defect, Math. Ann. 184 (1970), 197–214.

[20] K.K. Oberai, On the Weyl spectrum II, Illinois J. Math. 21, 84–90.

[21] K. Tanahashi, On log-hyponormal operators, Integral Equations Operator Theory, 34 (1999), 364–372.

(received 19.03.2007, in revised form 14.09.2007)

Department of Mathematics, Faculty of Science, University of Batna, 05000, Batna, Algeria.

E-mail: lombarkiafarida@yahoo.fr

Department of Mathematics, Faculty of Science, King Khalid University, Abha, P.O.Box 9004, Saudi Arabia.

E-mail: bachir ahmed@hotmail.com